Express $z_1=4-4i$ in polar form. Express your answer in exact terms, using degrees, where your angle is between $0^\circ$ and $360^\circ$, inclusive. $z_1=$
Solution: The Strategy A complex number in rectangular form, $z={a}+{b}i$, can be written in polar form as $z={r}[\cos{\theta}+i\sin{\theta}]$, where ${r}$ is the absolute value, or modulus, and ${\theta}$ is the angle, or argument. Therefore, ${r}$ and ${\theta}$ can be found using the following formulas: ${r}=\sqrt{{a}^2+{b}^2}$ $\tan{\theta}=\dfrac{{b}}{{a}}$ [How did we get these equations?] Similarly, a complex number in polar form, $z={r}[\cos{\theta}+i\sin{\theta}]$, can be written in rectangular form as $z={a}+{b}i$, using the following formulas: ${a}={r}\cos{\theta}$ ${b}={r}\sin{\theta}$ [How did we get these equations?] Finding $r$ For $z_1={4}{-4}i$ : ${a} = {4}$ ${b} = {-4}$ Therefore, we can find ${r}$ as follows. $\begin{aligned}{r}&=\sqrt{{a}^2+{b}^2} \\\\&=\sqrt{{4}^2+({-4})^2} \\\\&=\sqrt{16+16} \\\\&={\sqrt{32}} \\\\&={4\sqrt{2}}\end{aligned}$ Finding $\theta$ Using the formula, we have: $\begin{aligned}{\theta}&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{-4}}{{4}}\right) \\\\&={-45^\circ}\end{aligned}$ Since ${a}$ is positive and ${b}$ is negative, ${\theta}$ must lie in Quadrant $\text{IV}$. Therefore its angle must be between $270^\circ$ and $360^\circ$. Using the identity $\tan(360+\theta)=\tan(\theta)$, we know that the following is also a solution of the equation. $360^\circ+(-45^\circ)=315^\circ$ So $\theta = {315^{\circ}}$. Summary $z_1={4\sqrt{2}}[\cos{315^\circ}+i\sin{315^\circ}]$